Rogue peakon, well-posedness, ill-posedness and blow-up phenomenon for an integrable Camassa-Holm type equation

Authors

Mingxuan Zhu
Zhenteng Zeng
Zaihong Jiang
Baoqiang Xia
Zhijun Qiao, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

2023

Abstract

In this paper, we study an integrable Camassa-Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution - called {\sf rogue peakon}, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space Bsp,r with 1≤p,r≤∞ , s>max{1+1/p,3/2} or B3/22,1 , and then prove the ill-posedness of the solution in B3/22,∞ . Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if m0(x)=u0−u0xx≥(≢)0 , then the corresponding solution exists globally, meanwhile, if m0(x)≤(≢)0 , then the corresponding solution blows up in a finite time.

Recommended Citation

Zhu, Mingxuan, et al. "Rogue peakon, well-posedness, ill-posedness and blow-up phenomenon for an integrable Camassa-Holm type equation." arXiv preprint arXiv:2308.11508 (2023).